So the problem, that I have is the following:
Let $\{Y_t\}$ be a stationary process with $\mu=0$ and $a,b$ constant. Define and new stochastic process $X_t = (a + bt)s_t + Y_t$, where $s_t$ is the seasonal component. I need to prove that $\nabla \nabla_{12} X_t := (1 − B_{12})^2 X_t$ is stationary.
After solving I get that: $\nabla\nabla_{12} X_t= X_t - 2Xt_{-12} + X_{t-24}$ Then I substitute and I get:$E[X_t]= E[Y_t + 2Y_{t-12} + Y_{t-24}]$. Since $Y_t$ is stationary, then it follows that $E[X_t]$ also depends on t when I think about it but the $E[Y]$ is zero. So I am not sure if my solution for $E[X_t]$ is correct at all. Can some give me some idea if I am on the right track or I have totally messed up? Thanks a lot in advance :)
Since $(Y_t)$ is stationary, $E[Y_t]$ does not depend on $t$. Then you have a typo: the equality $E[X_t]= E[Y_t + 2Y_{t-12} + Y_{t-24}]$ should read $E[X_t]= E[Y_t - 2Y_{t-12} + Y_{t-24}]$ but you are right, all the terms there are zero hence $E[X_t]=0$.